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Evaluating Density Forecasts with an Application to Stock Market Returns

Density forecasts have become quite important in economics and finance. For example, such forecasts play a central role in modern financial risk management techniques like Value at Risk. This paper suggests a regression based density forecast evaluation framework as a simple alternative to other approaches. In simulation experiments and an empirical application to in- and out-of-sample one-step-ahead density forecasts of daily returns on the S&P 500, DAX and ATX stock market indices, the regression based evaluation strategy is compared with a recently proposed methodology based on likelihood ratio tests. It is demonstrated that misspecifications of forecasting models can be detected within the proposed regression framework. It is further demonstrated that the likelihood ratio methodology without additional misspecification tests has no power in many practical situations and therefore frequently selects incorrect forecasting models. The empirical results provide some evidence that GARCH-t models provide good density forecasts. The results further suggest that extensions of statistical models with fat-tailed conditional distributions to models that incorporate higher order conditional moments beyond the conditional variance might be appropriate to capture the empirical regularities in financial time series in some cases.

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Paper provided by Oesterreichische Nationalbank (Austrian Central Bank) in its series Working Papers with number 59.

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Length: 39
Date of creation: 18 Feb 2002
Date of revision:
Handle: RePEc:onb:oenbwp:59
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  1. Francis X. Diebold & Anthony S. Tay & Kenneth F. Wallis, 1998. "Evaluating Density Forecasts of Inflation: The Survey of Professional Forecasters," Working Papers 98-15, New York University, Leonard N. Stern School of Business, Department of Economics.
  2. Hansen, Bruce E, 1994. "Autoregressive Conditional Density Estimation," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 35(3), pages 705-30, August.
  3. Francis X. Diebold & Todd A. Gunther & Anthony S. Tay, 1997. "Evaluating Density Forecasts," Center for Financial Institutions Working Papers 97-37, Wharton School Center for Financial Institutions, University of Pennsylvania.
  4. Glosten, Lawrence R & Jagannathan, Ravi & Runkle, David E, 1993. " On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks," Journal of Finance, American Finance Association, vol. 48(5), pages 1779-1801, December.
  5. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
  6. Lumsdaine, Robin L, 1996. "Consistency and Asymptotic Normality of the Quasi-maximum Likelihood Estimator in IGARCH(1,1) and Covariance Stationary GARCH(1,1) Models," Econometrica, Econometric Society, vol. 64(3), pages 575-96, May.
  7. Clements, M.P. & Smith J., 1998. "Evaluating The Forecast of Densities of Linear and Non-Linear Models: Applications to Output Growth and Unemployment," The Warwick Economics Research Paper Series (TWERPS) 509, University of Warwick, Department of Economics.
  8. Francis X. Diebold & Jinyong Hahn & Anthony S. Tay, 1999. "Multivariate Density Forecast Evaluation And Calibration In Financial Risk Management: High-Frequency Returns On Foreign Exchange," The Review of Economics and Statistics, MIT Press, vol. 81(4), pages 661-673, November.
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